11.7 MATHEMATICAL MODELING WITH QUADRATIC FUNCTIONS. Objectives. Maximum Minimum Problems


 Aileen Estella Palmer
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1 a b Objectives Solve maimum minimum problems involving quadratic functions. Fit a quadratic function to a set of data to form a mathematical model, and solve related applied problems MATHEMATICAL MODELING WITH QUADRATIC FUNCTIONS We now consider some of the man situations in which quadratic functions can serve as mathematical models. a Maimum Minimum Problems We have seen that for an quadratic function f a b c, the value of f at the verte is either a maimum or a minimum, meaning that either all outputs are smaller than that value for a maimum or larger than that value for a minimum. (, f ()) a a (, f ()) f () at the verte a minimum f () at the verte a maimum There are man tpes of applied problems in which we want to find a maimum or minimum value of a quantit. If a quadratic function can be used as a model, we can find such maimums or minimums b finding coordinates of the verte. EXAMPLE 1 FencedIn Land. A farmer has 64 d of fencing. What are the dimensions of the largest rectangular pen that the farmer can enclose? 1. Familiarize. We first make a drawing and label it. We let l the length of the pen and w the width. Recall the following formulas: Perimeter: l w; Area: l w. 834 l w A w l To become familiar with the problem, let s choose some dimensions (shown at left) for which l w 64 and then calculate the corresponding areas. What choice of l and w will maimize A?
2 . Translate. We have two equations, one for perimeter and one for area: l w 64, A l w. Let s use them to epress A as a function of l or w, but not both. To epress A in terms of w, for eample, we solve for l in the first equation: l w 64 l 64 w l 3 w. Substituting 3 w for l, we get a quadratic function A w, or just A: A lw 3 w w 3w w w 3w. 3. Carr out. Note here that we are altering the third step of our fivestep problemsolving strateg to carr out some kind of mathematical manipulation, because we are going to find the verte rather than solve an equation. To do so, we complete the square as in Section 11.6: A w 3w 1 w 3w 1 w 3w w This is a parabola opening down, so a maimum eists. Factoring out ; We add, or FencedIn Land. A farmer has 1 d of fencing. What are the dimensions of the largest rectangular pen that the farmer can enclose? To familiarize ourself with the problem, complete the following table. l w A w 3w w Using the distributive law The verte is 16, 56. Thus the maimum value is 56. It occurs when w 16 and l 3 w Check. We note that 56 is larger than an of the values found in the Familiarize step. To be more certain, we could make more calculations. We leave this to the student. We can also use the graph of the function to check the maimum value (16, 56) Maimum: 56 A(w) (w 16) w 5. State. The largest rectangular pen that can be enclosed is 16 d b 16 d; that is, a square. Do Eercise 1. Answer on page A Mathematical Modeling with Quadratic Functions
3 calculator corner Maimum and Minimum Values We can use a graphing calculator to find the maimum or minimum value of a quadratic function. Consider the quadratic function in Eample 1, A w 3w. First, we replace w with and A with and graph the function in a window that displas the verte of the graph. We choose, 4,,3, with Xscl 5 and Yscl. Now, we press nd CALC 4 or nd CALC ENTER to select the MAXIMUM feature from the CALC menu. We are prompted to select a left bound for the maimum point. This means that we must choose an value that is to the left of the value of the point where the maimum occurs. This can be done b using the left and rightarrow kes to move the cursor to a point to the left of the maimum point or b keing in an appropriate value. Once this is done, we press ENTER. Now, we are prompted to select a right bound. We move the cursor to a point to the right of the maimum point or ke in an appropriate value. 3 3 Y1 X 3X 3 3 Y1 X 3X LeftBound? X Y Xscl 5 Yscl RightBound? X Xscl 5 Y Yscl We press ENTER again. Finall, we are prompted to guess the value at which the maimum occurs. We move the cursor close to the maimum or ke in an value. We press ENTER a third time and see that the maimum function value of 56 occurs when 16. (One or both coordinates of the maimum point might be approimations of the actual values, as shown with the value below, because of the method the calculator uses to find these values.) 3 3 Y1 X 3X 3 3 Guess? X Y Maimum X Y 56 4 nd To find a minimum value, we select item 3, minimum, from the CALC menu b pressing CALC ENTER. nd CALC 3 ENTER or Eercises: Use the maimum or minimum feature on a graphing calculator to find the maimum or minimum value of each function
4 b Fitting Quadratic Functions to Data As we move through our stud of mathematics, we develop a librar of functions. These functions can serve as models for man applications. Some of them are graphed below. We have not considered the cubic or quartic functions in detail other than in the Calculator Corners (we leave that discussion to a later course), but we show them here for reference. Linear function: f () m b Quadratic function: f () a b c, a Quadratic function: f () a b c, a Absolutevalue function: f () Cubic function: f () a 3 b c d, a Quartic function: f () a 4 b 3 c d e, a Now let s consider some realworld data. How can we decide which tpe of function might fit the data of a particular application? One simple wa is to graph the data and look for a pattern resembling one of the graphs above. For eample, data might be modeled b a linear function if the graph resembles a straight line. The data might be modeled b a quadratic function if the graph rises and then falls, or falls and then rises, in a curved manner resembling a parabola. For a quadratic, it might also just rise or fall in a curved manner as if following onl one part of the parabola Mathematical Modeling with Quadratic Functions
5 Choosing Models. For the scatterplots and graphs in Margin Eercises 5, determine which, if an, of the following functions might be used as a model for the data. Linear, f m b; Quadratic, f a b c, a ; Quadratic, f a b c, a ; Polnomial, neither quadratic nor linear. Sales Let s now use our librar of functions to see which, if an, might fit certain data situations. EXAMPLES Choosing Models. For the scatterplots and graphs below, determine which, if an, of the following functions might be used as a model for the data.. Population Linear, f m b; Quadratic, f a b c, a ; Quadratic, f a b c, a ; Polnomial, neither quadratic nor linear The data rise and then fall in a curved manner fitting a quadratic function f a b c, a. 3. Sales Population The data seem to fit a linear function f m b Sales Population The data rise in a manner fitting the right side of a quadratic function f a b c, a Life epectanc for women (in ears) Shoe size Source: Orthopedic Quarterl Answers on page A The data fall and then rise in a curved manner fitting a quadratic function f a b c, a.
6 6. U.S. Birth Rate for Women Ages Number of births per 1 women s Source: Centers for Disease Control and Prevention 198s 199s The data fall, then rise, then fall again. The do not appear to fit a linear or quadratic function but might fit a polnomial function that is neither quadratic nor linear. Do Eercises 5 on the preceding page. Whenever a quadratic function seems to fit a data situation, that function can be determined if at least three inputs and their outputs are known. EXAMPLE 7 River Depth. The drawing below shows the cross section of a river. Tpicall rivers are deepest in the middle, with the depth decreasing to at the edges. A hdrologist measures the depths D, in feet, of a river at distances, in feet, from one bank. The results are listed in the table at right. distance from left bank (in feet) D() depth of river (in feet) DISTANCE,, FROM THE RIVERBANK (in feet) DEPTH, D, OF THE RIVER (in feet) Mathematical Modeling with Quadratic Functions
7 6. Ticket Profits. Valle Communit College is presenting a pla. The profit P, in dollars, after das is given in the following table. (Profit can be negative when costs eceed revenue. See Section 3.8.) Profit P $ DAYS, PROFIT, P $ 1 $56 $87 $87 $548 $ 1 3 Das a) Make a scatterplot of the data. 4 5 b) Decide whether the data can be modeled b a quadratic function. c) Use the data points, 1, 18, 87, and 36, 548 to find a quadratic function that fits the data. d) Use the function to estimate the profits after 5 das. a) Make a scatterplot of the data. b) Decide whether the data seem to fit a quadratic function. c) Use the data points,, 5,, and 1, to find a quadratic function that fits the data. d) Use the function to estimate the depth of the river at 75 ft. a) The scatterplot is as follows. Depth (in feet) D b) The data seem to rise and fall in a manner similar to a quadratic function. The dashed black line in the graph represents a sample quadratic function of fit. Note that it ma not necessaril go through each point. c) We are looking for a quadratic function D a b c. We need to determine the constants a, b, and c. We use the three data points,, 5,, and 1, and substitute as follows: a b c, a 5 b 5 c, a 1 b 1 c. After simplifing, we see that we need to solve the sstem c,,5a 5b c, 1,a 1b c. Since c, the sstem reduces to a sstem of two equations in two variables:,5a 5b, (1) 1,a 1b. () We multipl equation (1) b, add, and solve for a (see Section 8.3): 4 5,a 1b, 1,a 1b 4 5a 4 5 a.8 a. River Depth Distance from the river bank (in feet) Adding Solving for a Answers on page A51 84
8 Net, we substitute.8 for a in equation () and solve for b: 1,.8 1b 8 1b 8 1b.8 b. This gives us the quadratic function: D.8.8. d) To find the depth 75 ft from the riverbank, we substitute: D At a distance of 75 ft from the riverbank, the depth of the river is 15 ft. 15 D().8.8 D(75) Do Eercise 6 on the preceding page. calculator corner Mathematical Modeling: Fitting a Quadratic Function to Data We can use the quadratic regression feature on a graphing calculator to fit a quadratic function to a set of data. The following table shows the average number of live births for women of various ages. AGE AVERAGE NUMBER OF LIVE BIRTHS PER 1 WOMEN a) Make a scatterplot of the data and verif that the data can be modeled with a quadratic function. b) Fit a quadratic function to the data using the quadratic regression feature on a graphing calculator. c) Use the function to estimate the average number of live births per 1 women of age and of age 3. (continued) Mathematical Modeling with Quadratic Functions
9 a) We enter the data on the STAT list editor screen, turn on Plot 1, and plot the points as described in the Calculator Corner on p The data points rise and then fall in a manner consistent with a quadratic function. L1 L L(7) 6.8 L3 b) To fit a quadratic function to the data, we press STAT 5 VARS 1 1 ENTER. The first three kestrokes select QuadReg from the STAT CALC menu and displa the coefficients a, b, and c of the regression equation a b c. The kestrokes VARS 1 1 cop the regression equation to the equationeditor screen as 1. We see that the regression equation is We can press ZOOM 9 to see the regression equation graphed with the data points. QuadReg a b c a b c Plot1 Plot Plot3 \Y X^ X \Y \Y3 \Y4 c) Since the equation is entered on the equationeditor screen, we can use a table set in ASK mode to estimate the average number of live births per 1 women of age and of age 3. We see that when, 85.8, and when 3, 11.8, so we estimate that there are about 85.8 live births per 1 women of age and about 11.8 live births per 1 women of age 3. 3 X Y X Remember to turn off the STAT PLOT as described on p. 548 before ou graph other equations. Eercise: 1. Consider the data in the table in Eample 7. a) Use a graphing calculator to make a scatterplot of the data. b) Use a graphing calculator to fit a quadratic function to the data. Compare this function with the one found in Eample 7. c) Graph the quadratic function with the scatterplot. d) Use the function found in part (b) to estimate the depth of the river 75 ft from the riverbank. Compare this estimate with the one found in Eample 7. 84
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